A Polynomial - Time Reduction from Bivariate to Univariate Integral Polynomial

A Abstract n algorithm is presented which reduces the prob-m lem of finding the irreducible factors of a bivariate polyno-ial with integer coefficients in polynomial time in the total i degree and the coefficient lengths to factoring a univariate nteger polynomial. Together with A. Lenstra's, H. Lenstra's u and L. Lovasz' polynomial-time factorization algorithm for nivariate integer polynomials and the author's multivariate-i to bivariate reduction the new algorithm implies the follow ng theorem. Factoring a polynomial with a fixed number of n b variables into irreducibles, except for the constant factors, ca e accomplished in time polynomial in the total degree and-e the size of its coefficients. The new algorithm can be gen ralized to reducing multivariate factorization directly to-m univariate factorization and to factoring multivariate polyno ials with coefficients in algebraic number fields and finite fields in polynomial time.